Anisotropic diffusions are classified by the second eigenvalue of the Hessian matrix associated with the diffusivity function into two categories: one incapable of edge-sharpening and the other capable of selective edge-sharpening. A third class is proposed: the eigenvalue starts with a small value and decreases monotonically with image gradient magnitude, so that the stronger the edge is, the more it is sharpened. Two families of such diffusivity functions are proposed. Numerical simulations indicate that the noise removal performance of anisotropic diffusion does not correlate with the shape of the diffusivity function, but is, instead, determined by the shape of the second eigenvalue function. Diffusivity functions in the third category produce the best maximum peak signal-to-noise ratio in numerical simulations.